Tits alternative for automorphism groups of compact Kähler manifolds
type: Item
instance of: scholarly article
title: Tits alternative for automorphism groups of compact Kähler manifolds
zbMATH Open document ID: 1271.14056
author: Tien-Cuong Dinh
published in: Acta Mathematica Vietnamica
publication date: 2013-01-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.3681
full work available at URL: Toc ACTA 4 37.htm
review text: This survey focuses on some properties of the automorphisms groups of compact Kähler manifolds. In particular, the main result proved is a result by \textit{J. Keum, K. Oguiso} and \textit{D.-Q. Zhang}, Math. Res. Lett. 16, No. 1, 133--148 (2009; Zbl 1172.14025)] and by \textit{D.-Q Zhang} [Invent. Math. 176, No. 3, 449--459 (2009; Zbl 1170.14029)]. Results by \textit{S. Cantat} [Ann. Math. (2) 174, No. 1, 299--340 (2011; Zbl 1233.14011)] and by Campana-Wang-Zhang are also reviewed. Let \(X\) be a compact Kähler manifold of dimension \(k\) and of Kodaira dimension \(\kappa_X\); take \(\kappa:=\max (\kappa_X,0)\) if \(\kappa_X>k\), \(\kappa:=k-1\) otherwise. Let \(G\) be a group of holomorphic automorphisms of \(X\) which does not contain any free non-abelian subgroup. Then it is shown that \(G\) admits a finite index subgroup \(G'\) such that: (i) \(G'\) is solvable, or equivalently, \(\mathrm{Aut}(X)\) satisfies the Tits alternative; (ii) the set \(N'\) of zero entropy elements of \(G'\) is a normal subgroup of \(G'\) and \(G'/N'\) is a free abelian group of rank at most \(k-\kappa-1\). All ideas necessary to the proof are reviewed in the first sections of the paper, avoiding technicalities: a mixed version of the Hodge-Riemann theorem, topological entropy and dynamical degrees and finally fibrations and the notion of relative dynamical degree. The last section contains the details of the proof of the main theorem cited above.
reviewed by: Chiara Camere
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6133190
zbMATH Keywords: tits alternative
zbMATH Keywords: virtually solvable group
zbMATH Keywords: topological entropy
zbMATH Keywords: dynamical degree
zbMATH Keywords: mixed Hodge-Riemann theorem
zbMATH Keywords: automorphism of Kähler manifold
MaRDI profile type: Publication
arXiv ID: 1210.3681
description: scientific article